What are the counterparts of non-linearities and dropout in fully convolutional networks?
All-convolutional neural network is a more general concept which can be (and is often) used without deconvolutional and unpolling layers, e.g. for an ordinary classification task. The idea is to replace the pooling and fully-connected layer with particular convolutional layers that do the same. Note that activation functions and dropout are not affected by this transformation. You can read more about it here.

The same will be true for the deconvolutional layers. I didn't find it stated explicitly in the original paper, but here's another work that applies fully-convolutional network for image segmentation. On the figure 2, you can see ReLu activations: they are placed after (simple) convolutional layers, in both upsampling and downsampling parts. They don't use dropout, but batchnorm instead, but if there were a dropout, just as before, it should be placed after the convolutional layer. The keep probability, though, should probably be higher - this relates to hyperparameters tuning and may be different depending on the problem and available data.